Schedule for: 25w5361 - Mixtures of Probability and Geometry

Systolic geometry studies relationships between geometry of the shortest non-contractible closed curves on non-simply connected Riemannian manifolds and other metric invariants. We will describe its connections to the inequalities that provide upper bounds for the widths of $M\subset B$ in terms of the volume or Hausdorff contents of $M$. The widths $W_m^B(M)$ measure how far $M$ is from a $m$-dimensional simplicial complex in $B$. Then we will talk about new inequality $W_{m-1}^{l^\infty}(M)\leq {\rm const} \sqrt{m}\ vol(M^m)^{1\over m}$ for closed manifolds $M^m\subset{\mathbb R}^N$ and its implications to systolic geometry. Here the width is measured with respect to the $l^\infty$ distance in the ambient Euclidean space. This inequality is a joint result with Sergey Avvakumov. Its proof involves a version of kinematic formula where one averages over isometries of $l^N_\infty$, and a probabilistic construction of high-codimension analogs of optimal foams recently discovered by Kindler, O'Donnel, Rao, and Wigderson and used in theoretical computer science. Then we will discuss more general width-volume and width-Hausdorff content inequalities, their connections with the isoperimetric inequality for Hausdorff contents jointly proven by Y.Liokumovich, B. Lishak, the speaker, and R. Rotman, and the boxing inequalities proven by S. Avvakumov and the speaker. (Recall that for each $m$ the $m$-dimensional Hausdorff content ${\rm HC}_m(X)$ is defined as the infimum of $\Sigma_i r_i^m$ over all coverings of $X$ by metric balls, where $r_i$ denote the radii of these balls.) \end{document}

Percolation theory traditionally studies the connectivity properties of various types of random graphs in a geometric setting. Although many of these models can be naturally generalized from graphs to cell complexes, it can be difficult to find the right questions to ask in this higher-dimensional setting, much less the answers. We will give an introduction to a topological view of these complexes, in which an analogue of the classical phase transition appears in the homology of the space.

In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by $n \geq 2$ distinct landmark points in $\mathbb{R}^d$. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold $Q$ of $n$ distinct landmark points in $\mathbb{R}^d$ can be endowed with a Riemannian metric $g$ such that the above optimisation problem is equivalent to the geodesic boundary value problem for $g$ on $Q$. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold $(Q,g)$ is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in $\mathbb{R}^d$ and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of $(Q,g)$.

For decades, the study of the spectrum of the Laplacian of Riemannian manifolds has been a very active topic of research at the crossroads of Geometry, Dynamics, Number Theory and Probability. The particularly rich and beautiful theory for hyperbolic surfaces (i.e. with constant curvature -1) holds a privileged spot in the area because it deals with objects that are explicit enough to allow us to get our hands on, yet it still holds many mysteries. One of the broad goals of the area is to understand the behaviour of the Laplace eigenvalues of a "typical" hyperbolic surface. In this talk I will present a spectral gap result for random hyperbolic surfaces of infinite area without cusps (aka Schottky surfaces), obtained in collaboration with M. Magee and F. Naud. Our result can be interpreted as a probabilistic analog for Schottky surfaces of Selberg's celebrated 1/4-Conjecture.

Construct a random knot by selecting a sequence of $N$ uniformly random points in the cube and connecting them in cyclic order. This model was proposed by Millett for studying polymer molecules in a constrained volume. What is the expected area of the minimal area Seifert surface of this knot? The answer turns out to be on the order of $\sqrt{N \log N}$. This one tangible result (in reality more general than stated here) is at the crossroads of two much vaguer stories. One: this model should be part of a universality class which I haven't yet been able to formulate. Two: I stumbled upon it while trying to investigate the topology of random maps from the 3-sphere to the 2-sphere, but I'm no longer convinced that question is well-posed. On these two topics -- isoperimetry of random cycles and topology of random maps -- I'd like to invite participants to help formulate the right definitions and the right questions.

Topology allows us to move beyond visible shapes to uncover hidden structures in data. Through Topological Data Analysis (TDA) we can identify patterns and relationships in complex biological systems, offering a new perspective on their organization. This tool has proven valuable in studying horizontal gene transfer and the evolution of microbial communities. In this talk, we will explore two applications. The first focuses on non-vertical gene inheritance. We show how persistent homology can detect horizontal gene transfer (HGT) by revealing deviations from vertical inheritance in both simulated populations and two real datasets. Starting from hierarchical processes with random mutations, we find that the appearance of 1-holes in barcodes reflects non-tree-like structures generated by HGT events. The second application focuses on classifying multicellular patterns. We investigate how comparing persistence diagrams using different metrics enables us to distinguish cellular arrangements in a simulated dataset with two cell types, as well as in a dataset of cells at various stages of cancer. The first project is joint work with S. Guerrero-Flores and N. Sélem-Mojica. The second is ongoing work in collaboration with L.R. Figueroa-Martínez, E. Azpeitia, P. Siliceo, and A. Rodriguez.

The cops and robbers game is a well-studied graph theory model, where cops try to corner a robber on a graph. The cop number is the smallest number of cops needed to deterministically capture a robber. The zombies are random versions of the cops: one survivor and $M$ zombies take turns moving on a graph. The zombies pick a random edge to move towards the survivor. We prove that on the $n x n$ torus, it is possible to escape from $n/\log(n)$ zombies, but not from $n^{3/2}$ zombies. Joint work with Pawel Pralat and Peter Winkler.